Page 118 - Vector Analysis

114 CHAPTER 4. Vector Calculus

f ˝ ψ : V Ñ Rn is differentiable at ψ´1(p). The derivative of f at p, denoted by dfp, is a
linear map on Tp Σ satisfying

(dfp)(v) = dˇ ˝ x)(t) ,
ˇ (f

dt ˇt=0

where x : (´δ, δ) Ñ Σ is a C 1-parametrization of a curve on Σ such that x(0) = p and

x 1(0) = v. A scalar function f : Σ Ñ R is said to be of class C 1 if f ˝ ψ is of class C 1 for

all local parametrization tV, ψu.

4.3.2 The metric tensor and the first fundamental form

Definition 4.41 (Metric). Let Σ Ď R3 be a regular surface. The metric tensor associated
with the local parametrization tV, ψu (at p P Σ) is the matrix g = [gαβ]2ˆ2 given by

gαβ = ψ,α ¨ψ,β = 3 B ψi B ψi in V

ÿ

i=1 B yα B yβ

or equivalently, g = [Dψ]T[Dψ].

Proposition 4.42. Let Σ Ď R3 be a regular surface, and g = [gαβ]2ˆ2 be the metric tensor
associated with the local parametrization tV, ψu (at p P Σ). Then the metric tensor g is

positive definite; that is,

2 @v = 2 Bψ ‰ 0.
B yγ
ÿ gαβvαvβ ą 0 ÿ vγ

α,β=1 γ=1

Proof. Since Dψ has full rank on V, every tangent vector v can be expressed as the linear

combination of !Bψ , B ψ ) Write v = 2 vγ Bψ . Then if v ‰ 0,
. B yγ
B y1 B y2 ř

γ=1

0 ă }v}2R3 = 3 2 vα Bψi vβ Bψi = 2

ÿ ÿ ÿ gαβvαvβ .

i=1 α,β=1 B yα B ψβ α,β=1 ˝

Definition 4.43 (The first fundamental form). Let Σ Ď R3 be a regular surface, and
g = [gαβ]2ˆ2 be the metric tensor associated with the local parametrization tV, ψu (at
p P Σ). The first fundamental form associated with the local parametrization tV, ψu (at
p P Σ) is the scalar function g = det(g).

Theorem 4.44. Let Σ Ď R3 be a regular surface, and tV, ψu be a local parametrization at

p P Σ. Then ?g = }ψ,1 ˆψ,2 }R3 .

(4.8)